I'm fascinated by this kind of "what if" but I'm pretty sure they don't make sense.
Fact is, it's no sense to say "the Romans" or "the Americans", 'cause tactics (and EVERYTHING else) change so quickly that you should state not only an era but maybe a single day!
Examples: in the earliest part of the 100 Years war, Englishmen defeated the French cavalry by simply standing still and throwing arrows at them. This could happen 'cause the French used "trad" war system consisting in courageously charging (even up-hill!)
When the guidance of the war on the French side was taken by the intelligent (even if less "noble") Grand Constable, he simply ordered to fight in a more prudent way and this changed the war.
Romans: which ones? Those of the Republican or Ceasarian era? Or those of the late Imperial era? They were different people (the former were conscript citizens, the latter mostly foreign mercenaries), differently armed (shields became smaller, swords longer, etc.), using different tactics (more cavalry, less infantry), with different moral.